AN HOURLY DIFFUSE FRACTION MODEL WITH CORRECTION FOR
VARIABILITY AND SURFACE ALBEDO
by
Arvid Skartveit ', Jan Asle Olseth ' # and Marit Elisabet Tuft #
Geophysical Institute, University of Bergen
AllJgaten 70, N-5007 Bergen, NORWAY
' ISES members
# also affiliated at the Norwegian Meteorological Institute, Bergen
(Revised version, accepted for publication in Solar Energy)
Pages 1 - 11, Figs. 1 - 8, Tables 1 -3
ABSTRACT
The paper presents an improved version of a previously published model for the diffuse fraction of hourly
global irradiance. In addition to hourly solar elevation and clearness index, an hour-to-hour variability
index and regional surface albedo are included among the input parameters. Moreover, to prevent
excessively high normal incidence beam irradiances at very low solar elevations, the model does not allow
a solar elevation dependent maximum beam transmittance to be exceeded. This new model is tuned to 32
years of data from Bergen, Norway. Moreover, a test against independent data from 4 European stations
showed that the model performs better than the models of Erbs et al. (1982), Maxwell (1987), and Perez
et al. (1992).
1. INTRODUCTION
Observations of solar radiation are most frequently of global radiation, while irradiation on sloping
surfaces is needed for a number of applications. The division of global radiation into its beam and diffuse
components is therefore frequently needed to derive requested data from available data.
A number of diffuse fraction models are available for averaging times ranging from one month to one
hour or even less. The present paper focuses on hourly data, for which time scale several comparisons
between diffuse fraction models are reported. Davies et al. (1988) found, from test against a
comprehensive set of experimental data, that the model of Erbs et al. (1982) performed best among a
number of models. From test against another extensive data set, Perez et al. (1990a) found that the quasiphysical model of Maxwell (1987) performs better overall than does the model of Erbs et al. (1982) and
that of Skartveit and Olseth (1987). According to Olmo et al. (1996), a total of 8 models tested against
data from Almeria (Spain) tended to underestimate the diffuse fraction in the wake of the Mt. Pinatubo
eruption.
Different diffuse fraction models require in general different input data. Thus, the model of Erbs et al.
(1982) requires only hourly clearness index as input. The Maxwell (1987) model and our old model
(Skartveit and Olseth, 1987) require hourly clearness index and solar elevation as input. Perez et al.
(1992) add even ambient dew-point temperature and an hour-to-hour variability index to the list of input
data. From a large multiclimatic data base they derive a computationally effective model which involves a
four-dimensional look-up table consisting of a 6X6X5X7 matrix of numerical constants. In the present
paper, the Perez model is run with climatological monthly averages of dew point temperature.
With particular reference to our old model (Skartveit and Olseth, 1987), which according to Olmo et al.
(1996) and Dumortier (1997) tend to overestimate the diffuse fraction under cloudless sky, the present
paper will focus on the model improvement obtained by adding new parameters (hour-to-hour variability
index and regional surface albedo) to the list of input data (Tuft, 1993). Moreover, in order to obtain a
more condensed format and to ease our understanding of the diffuse/beam partitioning of global
irradiance, we prefer to phrase our model in terms of a limited number of analytical expressions rather
than resorting to extensive look-up tables.
2. DATA
Our new model is tuned to data measured at Bergen (60o24'N, 5o19'E, 45 m above m.s.l), Norway. This
developmental data sample consists of 32 years (1965-96) of hourly global and diffuse irradiance. The
data are described by Skartveit & Olseth (1987) and Radiation Yearbooks (1965-96). Only hours during
which the sun is completely above the somewhat elevated horizon are included in the analysis. Moreover,
only data for the season April to October are used in order to avoid cases with significant snow cover.
Global and diffuse horizontal irradiances are measured by Eppley / Kipp&Zonen pyranometers. The
sensor for diffuse irradiance is shaded with a 6 cm diameter shading disk at 30 cm distance. For the period
1991-96, normal incidence beam irradiance is measured by an Eppley Normal Incidence Pyrheliometer
2
(NIP).
Moreover, from a total of 11598 days of data from Bergen, 103 cloudless days were carefully selected
according to criteria involving visual cloud observations and forenoon-afternoon irradiance symmetry
(Olseth and Skartveit, 1989). Note that these hours are cloudless to a degree substantially beyond any
hours selected solely from globalirradiance data.
A few selected models are tested against independent verification samples of data measured at the four
European stations of the International Daylight Measuring Programme (IDMP): Lisbon, Lyon, Garston,
and G@vle (Dumortier et al., 1994). Data from the General class stations at Lisbon (38o 46'N, 9o 08'W,
106 m above m.s.l.; January 1993 - June 1994), Lyon (45o47'N, 4o56'E, 170 m above m.s.l.; Year 1994),
and the Research class station Garston (51o43'N, 0o22'W, 80 m above m.s.l.; Year 1992), were gratefully
received from D. Dumortier at ENTPE, Lyon. Besides, data from the Swedish General class station G@vle
(60o40'N, 17o10'E, 16 m above m.s.l.; Year 1995) were gratefully received from H. A. L`fberg at the
Royal Institute of Technology, Department of Built Environment.
Global and diffuse horizontal irradiances are measured by Kipp&Zonen CM6 pyranometers at Lisbon and
Lyon, and by Kipp&Zonen CM11 pyranometers at Garston and G@vle. The sensors for diffuse irradiance
are shaded with shadow rings at Lisbon (7.7 cm width, 31.5 cm radius), Lyon (5 cm width, 17.8 cm
radius),Garston (5 cm width, 25.4 cm radius) and with shading disk (7 cm radius, 70 cm distance) at
G@vle. For Lisbon, Lyon, and Garston the diffuse irradiance is corrected with the shadowband correction
of Littlefair (1989).
3. THE NEW DIFFUSE FRACTION MODEL
3.1 The variability index
Clouds change the diffuse fraction by affecting diffuse and beam irradiance in a multiplicity of ways. For
example, while breaks in an extensive cloud deck may primarily enhance the beam irradiance, scattered
clouds on the sky dome may enhance the diffuse irradiance and leave the beam irradiance unaffected. As a
rough diagnostic tool to statistically detect the presence of such variable/inhomogeneous clouds, Skartveit
and Olseth (1992) introduced, in a similar manner as Perez et al. (1992), an hourly variability index F3. It
is defined as the root mean squared deviation between the Oclear skyO index of the hour in question (D)
and, respectively, the preceding (D-1) and the deceding (D+1) hour:
σ3 = { [ ( ρ - ρ-1 )2 + ( ρ - ρ+1 )2 ] / 2 }0.5 ,
( 1a )
σ3 = | ρ - ρ±1 | ,
( 1b )
where the latter expression is used when either the preceding or the deceding hour is missing. Moreover,
clear sky index D and clearness index k are defined as follows:
ρ = k / (k1) ,
( 2a )
k = Hg / Hex ,
( 2b )
where Hg and Hex are surface global and extraterrestrial global irradiance, while k1 is a measure of the
cloudless clearness index, given by eqn (6b) below.
The above F3 is nearly independent of solar elevation, while it varies markedly with clear sky index D (Fig.
1a). Not unexpectedly, it appears that low F3 is associated with overcast (D . 0) or nearly cloudless (0.9 <
D < 1.0) sky, while high F3 is associated with broken clouds (intermediate or even extremely high D). For
our snow-free data, extremely high D is most probably due to the combination of clouds and unobscured
sun. Note, however, that cloud effects are significant even within the nearly cloudless bin (0.9 < D < 1.0).
Within this bin, the F3 averages for all hours (Fig. 1a) exceed the F3 averages for the subset of hours
3
which are known to be cloud-free from visual cloud observations (Fig. 1b). In fact, some 5% of the F3
values within the 0.9 - 1.0 bin do exceed 0.3. If F3 for any reason is unknown, the following average F3
may be applied:
σ3 = 0.021 + 0.397 ρ - 0.231 ρ2 - 0.13 exp [ - { [ ( ρ - 0.931 ) / 0.134 ]2 }0.834 ] ,
( 3a )
σ3 = 0.12 + 0.65 ( ρ - 1.04) ,
( 3b )
for D # 1.04 (eqn 3a) and for D $ 1.04 (eqn 3b). This average F3, which is plotted as a function of clear
sky index D in Fig. 1, is found by least squares analysis. Note that 1.04 is found to be the limit which
yields the minimum overall residual error relative to eqns. (3a-b), and it is not chosen for reasons beyond
that empirical fact.
3.2 Snow-free environment
The diffuse fraction model is tuned by least squares analysis to data from Bergen (1965 - 96, 77 479
hours) for the months April - October, during which season a regional (snow-free) surface albedo of 0.15
is a realistic estimate.
Invariable hours
For F3 . 0, the diffuse fraction d (diffuse / global) within four bins I-IV of clearness index k, is found to
be:
I) For k # 0.22, no significant beam irradiance is observed at any solar elevation:
d = 1.00 .
(4)
II) For 0.22 # k # k2, broken clouds which partly obscure the sun, prevail. We find:
d = f ( k, h ) = 1- (1 - d1) ( 0.11 √K + 0.15 K + 0.74 K2) ,
(5)
K = 0.5 { 1 + sin [π
π (k - 0.22)/ (k1 - 0.22 ) - π/2 ] } ,
( 6a )
where
k1 = 0.83 - 0.56 exp ( -0.06 h ) ,
k2 = 0.95 k1 ,
d1 = 0.07 + 0.046 (90 - h) / ( h + 3 ) ,
( 6b )
( 6c )
( 6d )
and where h is solar elevation in degrees, and d1 is put equal to unity for h# 1.4o.
III) For k2 # k # kmax, where cloudless (or nearly so) skies prevail, we assume that the diffuse irradiance
is a fixed, but solar elevation and turbidity dependent fraction of the irradiance removed from the beam.
From this assumption we obtain:
d = d2 k2 ( 1 - k ) / [ k ( 1 - k2 ) ] ,
(7)
where d2 = f ( k2 , h ) is calculated from eqns (5-6) above. The upper bin limit kmax is derived from the
assumption that the beam transmittance does not exceed a maximum value bkmax:
kmax = [ kb max + d2 k2 /( 1 - k2 )] / [ 1 + d2 k2 /( 1 - k2 ) ] ,
(8)
where kb max is fitted to an OextremeO beam transmittance modelled under an Arctic Winter Atmosphere
(precipitable water vapour = 0.216 cm and 0.5 :m aerosol optical depth = 0.02) by the model SMARTS2
4
(Gueymard 1993):
kb max = 0.81α ,
( 9a )
α = [ 1/sin (h) ] 0.6 .
( 9b )
IV) For k $ kmax, we assume that any clearness index increase beyond kmax arises from diffuse irradiance
from clouds not obscuring the sun, while the beam transmittance remains fixed at the abovebkmax:
d = 1 - kmax ( 1 - dmax ) / k ,
( 10 )
where the diffuse fraction for clearness index kmax is obtained from eqn (7):
dmax = d2 k2 ( 1 - kmax ) / [ kmax ( 1 - k2 ) ] .
( 11 )
Diffuse fractions predicted for invariable hours by the above model are plotted against clearness index for
six solar elevations in Fig. 2.
Variable hours
For F3 > 0, the least squares analysis shows that a term ) ( k, h, F3 ) should be added to the above
invariability expressions (4), (5), (7), and (10) to account for the effect of variable/inhomogeneous clouds:
∆ ( k, h, σ3 ) = -3 kL2 (1 - kL ) σ31.3, for 0.14 ≤ k ≤ kx ,
∆ ( k, h, σ3 ) = 3 kR (1 - kR )2 σ30.6, for kx ≤ k ≤ ( kx + 0.71 ) ,
∆ ( k, h, σ3 ) = 0 for k ≤ 0.14 and for k ≥ kx + 0.71 ,
(12a )
(12b )
(12c )
where we have applied the following linear transformations of clearness index k:
kx = 0.56 - 0.32 exp ( - 0.06 h) ,
kL = ( k - 0.14 ) / ( kx - 0.14 ) ,
kR = ( k - kx ) / 0.71 .
(13a )
(13b )
(13c )
As expected, these empirical equations reflect that increasing F3 is correlated with decreasing diffuse
fraction for low k (k < kx), and with increasing fraction for k > kx (Fig. 3). Note that the sensitivity to F3 is
particularly high at high clearness indices, where the diffuse fraction increases by a factor 2 - 5 when F3
increases from 0.0 to 0.3. This variability is quite important to account for in e.g. simulation of diffuse
dayligthing through windows or prediction of the performance of high threshold focusing solar systems.
3.3 Surface albedo and diffuse fraction
The above diffuse fraction model is a "snow-free terrestrial" model. That is, it is tuned to measurements in
a snow-free terrestrial environment with surface albedo r* . 0.15. If a clearness index kr is measured at
solar elevation h in an environment with surface albedo r deviating significantly from this r*, the diffuse
irradiance is affected by multiple reflections between the surface and the sky dome. These changes need to
be corrected for, and we propose a correction procedure based on the following three assumptions:
1) For zero surface albedo, the atmosphere absorbs a fraction A of the solar irradiance impinging
on its top. This fraction is fairly independent of both solar elevation and cloudiness, and A = 0.20
may be taken as a rough average (Paltridge & Platt, 1976).
2) The atmospheric albedo R is the same to reflected (diffuse) irradiance from below as it is to
beam irradiance from the sun at elevation h' = 37o, in accordance with the commonly used
"diffusivity factor" 1/sin(h') = 1.66 (Paltridge & Platt, 1976).
5
3) The clearness index kr, measured at a single point, is (in an average sense) close to the
2
corresponding index averaged over a surrounding region of some few hundred km
.
Under these assumptions, energy conservation requires:
1 = k0' + A + R ,
( 14 )
where k0' is the clearness index for zero surface albedo at solar elevation h' under the actual sky. Assuming
multiple reflections between the surface and the actual sky dome, we may derive k0' from the
corresponding clearness index kr' for actual surface albedo r. Under the assumption of fixed clear sky
index D, kr' may in turn be derived from the actual clearness index rk:
k0' = kr' (1 - r R) ,
( 15 )
kr' = kr k1(h') / k1(h) ,
( 16 )
where the cloudless indices k1(h') and k1(h) are given by eqn (6). The above equations may be solved for
the highly cloud dependent atmospheric albedo R:
R = (1 - A - kr') / (1 - kr' r) ,
( 17 )
where any R < 0.08 is replaced by an approximate minimum value 0.08 (Gueymard, 1993), since the
above assumption 3) breaks down in those rare cases when high clearness indices are due to high diffuse
irradiance from clouds not obscuring the sun. With this estimate of R, we may now calculate the clearness
index kr* for surface albedo r* at actual solar elevation under the actual sky:
kr* = kr(1 - r R) / (1 - r* R) .
( 18 )
With this kr* as input, the diffuse fraction dr* for surface albedo r* can now be calculated from our "snowfree terrestrial" diffuse fraction model, and the corresponding beam irradiance can be obtained. This beam
irradiance, being independent of surface albedo, can finally be subtracted from the actual global irradiance
to yield actual diffuse irradiance and actual diffuse fraction:
dr = 1 - kr* (1 - dr*) / kr .
( 19 )
Here kr is the clearness index measured in the actual environment, kr* is obtained from eqn (18), while dr*
is obtained from our "snow-free terrestrial" diffuse fraction model with rk* as input.
Diffuse fractions from our model are plotted in Fig. 4 against clearness index for solar elevation 10 and
60E, and for surface albedo 0.15 and 0.6. Note, in particular, that the transformation of an 0.15 curve into
an 0.6 curve involves the assumption that the clearness index increase is due solely to increased diffuse
irradiance. We therefore have no change to those parts of the curves (clearness index bins I and IV) where
this assumption is implicit in the first place.
However, since the necessary surface albedo is extremely difficult to establish reliably for the surrounding
few hundred km2 of hilly inhomogeneous terrain, we have not verified the above correction procedure for
the surface albedo effect. We nevertheless feel confident that, if our model is to be used in cases where
regional albedo deviates significantly from 0.15, the output will be improved by applying the above
albedo correction.
Finally, note that any diffuse fraction d > 1 yielded by any of the preceding equations should be replaced
by 1.
4. RESULTS AND DISCUSSION
4.1 Cloudless skies
6
As mentioned above, 103 cloudless days at Bergen were carefully selected according to criteria involving
visual cloud observations and forenoon-afternoon irradiance symmetry. Among these cloudless hours, no
individual hour is in fact found in clearness index bin IV (k > kmax), in accordance with our assumption
that such hours are due to the combination of clouds and unobscured sun.
Within five solar elevation bins, the average diffuse fractions observed during these cloudless hours are
significantly lower than those predicted by both of our two models. These apparent model discrepancies,
relative to observations from the same data sample as to which both our models are tuned, derive from the
fact that none of the models are designed to predict diffuse fractions from explicit cloud information, but
rather from solar elevation, clearness index, and (in the case of our new model) variability index F3. Thus,
our old model does not predict the diffuse fraction for cloudless hours explicitly, but rather predicts the
average fraction for all hours with given solar elevation and clearness index. Due to the presence of
some clouds, even for the vast majority of hours with high clearness index, such predicted overall averages
turn out to exceed those observed under cloudless sky by an average amount 0.08 - 0.12 (Fig. 5).
Likewise, our new model is tuned to predict average diffuse fractions for all hours with given solar
elevation, clearness index and variability index. The fractions predicted by this model exceed those
observed under cloudless sky by an average amount now reduced to 0.03 - 0.06 (Fig. 5). This discrepancy
reduction means that the addition of variability index to the input list enables the model to partly account
for the presence of clouds during hours of high clearness index. The discrepancy is not entirely removed,
however, implying that even hours of low F3 and high clearness index are more or less affected by clouds.
Although these clouds are too minor/invariable to yield a "detectable" F3 increase, they nevertheless
increase the diffuse fraction significantly beyond that corresponding to an entirely cloud-free sky.
It turns out, however, that within the five solar elevation bins the observed average cloudless diffuse
fractions are only 0.01 - 0.02 lower than those from our new model run with zero F3 (Fig. 5), implying that
with zero F3 as input our model is fairly adequate for cloud-free sky. Moreover, within the four highest
solar elevation bins, the observed average fractions are within 0.01 from those yielded by the cloudless
model of Camps and Soler (1992), which is tuned to observations at Barcelona, Spain. Finally, in
accordance with the findings of Olmo et al. (1996), the average cloudless diffuse fractions at Bergen were
found to be higher during the 2 2 years following the Mt. Pinatubo eruption than prior / subsequent to
this event. This implies that turbidity information has the potential of improving the performance of a
diffuse fraction model.
4.2 All-weather skies
Observed group mean diffuse fractions for nearly invariable hours (σ3 < 0.03) nicely fit our new model, as
exemplified in Fig. 6 by three solar elevation bins of data from our verification sample (Lisbon, Lyon,
Gävle, and Garston).
An overall comparison of the Erbs et al. (1982), the Maxwell (1987), the Perez et al. (1992) model, along
with our old (Skartveit and Olseth, 1987) and our new model is given in Tables 1 - 3. The data from the 4
IDMP stations serve as an independent verification sample for all five models, while for the Bergen data
such independency is not the case for our old and our new model since they are tuned to these data.
However, independent of any model and relating only to the consistency between global, beam and diffuse
irradiance data, it should be noted that for the 6 years subset of Bergen data, the average normal incidence
beam irradiance (236 Wm-2) observed with Eppley NIP at solar elevation > 10o is identical to that
obtained from the observed difference between global and diffuse irradiances (Table 3).
For the total IDMP data sample, all root mean square deviations (RMSD) are lowest for our new model
and second lowest for the Perez model. This is noteworthy, since our model requires only solar elevation,
global irradiance, and the variability index as input, while the Perez model in addition requires dew point
temperature, which can not be derived from global radiation time series only. For our model, the root
7
mean square deviations (RMSD) for solar elevation h > 10o are 0.094 for diffuse fraction, 36 Wm-2 (23%
of the average) for diffuse irradiance, and 65 Wm-2 (20%) for normal incidence beam irradiance. The
above model ranking applies even to Bergen data, with the exception of h < 10o where RMSDs for our
new model slightly exceed those of the Perez model. However, since the Perez model requires dew point
temperature as input and the other models do not, it remains an open question whether the above
comparison would change slightly in favour of the Perez model if hourly dew point temperatures, rather
than monthly averages, were used as input for the verification sample. Finally, concerning computational
efficiency, the computation time for 75269 hours turned out to be negligibly larger (3%) for the Perez
model than for our new model.
For h > 10o, all mean bias deviations (MBD) are highest for the Maxwell model and our old model, both
for IDMP data and for Bergen data. For h < 10o , we see that MBDs relative to Eppley NIP data (Bergen)
are highest for the Erbs and Maxwell models. Relative to the total IDMP data sample, the mean bias
deviations of our new model are remarkably low for h > 10o : 0.009 for diffuse fraction, 3 Wm-2 (2%) for
diffuse irradiance, and -3Wm-2 (-1%) for normal incidence beam irradiance. Note, however, that the above
verification mean bias deviations are almost identical to those of the 6 years (1991-96) subset of Bergen
data, indicating that the minor mean bias may well be due to a minor temporal variation rather than to a
spatial contrast between sites. Note also that the model vs. observation conformity varies somewhat
between the stations. Thus, all models tend to overestimate the diffuse fraction at Lyon and underestimate
it at Gävle. Although the applied shadow-band correction accounts for the geometry in question, this
pattern may nevertheless be due to the fact that among our stations the solid angle blocked by the shading
device is lowest at Gävle and highest at Lyon.
Finally, to describe the model deviation pattern in some more detail, we group our verification data sample
according to solar elevation, clearness index and variability index. For high clearness index bins, the
observed group mean diffuse fractions are plotted against modelled counterparts in Fig. 7. The model
improvement obtained by adding the variability index to the input list (exemplified here by our new
model) is clearly demonstrated for all solar elevations. For the overall verification sample, the diffuse and
beam irradiance RMSDs of our new model are 82-84% of the average RMSD for the three models with no
variability correction, while for clearness index > 0.7 the corresponding percentage is reduced to 70%
(Tables 2-3). Maximum model improvement is thus obtained at high clearness index, in which case the
effect of scattered clouds is accounted for by means of information extracted from hourly global
irradiances only.
4.3 Low solar elevations
Since horizontal beam irradiance has to be divided by sin (h) to obtain normal incidence beam irradiance,
diffuse fraction models may produce substantial errors in normal incidence beam irradiance at low solar
elevations h, unless explicit care is taken, as done in both our old and our new model. To check that
particular aspect of our new model, we compare its output at hourly solar elevation < 10o to Eppley NIP
data from Bergen. This comparison is limited to hours without any horizon obstruction of the solar beam,
and these NIP data are not included in our developmental sample. The observed NIP irradiances conform
nicely both with those derived from the observed difference between global and diffuse irradiance (Fig.
8a) and with those yielded by our new diffuse fraction model from observed global irradiance (Fig. 8b).
The highest beam irradiances yielded by Eppley NIP and by our new diffuse fraction model, both slightly
exceed MODTRAN (Berk et al, 1989) irradiances under a Sub Arctic Summer atmosphere with visual
range 200 km, corresponding to 0.5 :m aerosol optical depth 0.06. However, no modelled or observed
irradiance in Fig. 8 exceeds the MODTRAN irradiance under an aerosol-free Sub Arctic Winter (SAW)
atmosphere. It can thus be concluded that our new diffuse fraction model, developed from horizontal
irradiances, even at low solar elevations conforms nicely both with measured (NIP) maximum beam
irradiances and with modelled (MODTRAN) maximum values.
8
However, the model of Erbs et al. (1982), which is chosen to exemplify a model without any explicit
upper limit for normal incidence beam irradiance, yields three values exceeding the MODTRAN
irradiance under an aerosol-free SAW atmosphere. In these cases, which are plotted as "E" in Fig. 8b, both
observations and values modelled with our new model and even with the three remaining models (not
shown on the figure) were significantly lower and more realistic. Note, however, that these three values
makes only 2% of the 147 cases with observed NIP irradiance > 120 Wm-2 (= WMO threshold for Obright
sunshineO). The important message is, however, that at solar elevation even closer to zero such lack of an
upper limit may cause diffuse fraction models to yield ridiculously high beam irradiances due to e.g. minor
measurement errors in global irradiance.
Even for the independent verification sample (IDMP) of horizontal irradiances (h < 10o), the mean bias
deviations of our new model are moderate: 0.031 for diffuse fraction, -1 Wm-2 for diffuse irradiance, and 3Wm-2 for normal incidence beam irradiance (Tables 1-3). The corresponding root mean square deviations
are 0.165, 8 Wm-2 (24%), and 74 Wm-2 (58%), i.e. quite significant on a percentage basis.
5. CONCLUDING REMARKS
The test of any hourly diffuse fraction model should include the comparison of model derived and
independently observed normal incidence beam irradiance. A such comparison is hardly the most
flattering one, but it is well designed to disclose important and potentially substantial model errors at low
solar elevations.
Test against independent data demonstrates that accounting for global radiation variability improves the
performance of a previously published diffuse fraction model. Relative to an independent verification
sample, the root-mean square deviation of hourly normal incidence beam irradiance thus decreases by
some 10-15%, to 65 Wm-2 at solar elevation h > 10o and to 74 Wm-2 at h < 10o. Maximum model
improvement is obtained at high clearness index, where RMSD is reduced by some 25-30%. This model
improvement, obtained by means of information extracted solely from hourly global irradiances, is
important to account for in e.g. simulation of diffuse dayligthing through windows or prediction of the
performance of high threshold focusing solar systems.
The overall mean bias deviation of normal incidence beam irradiance is remarkably small (-3 Wm-2), and
the mean bias deviations are moderate even for data binned according to solar elevation, clearness index
and variability index. The maximum modelled normal incidence beam irradiances conform nicely, even at
low solar elevations, both with observed (Eppley NIP) maxima and with maximum values modelled by
MODTRAN.
Due to multiple reflections between the surface and the sky dome, regional surface albedo significantly
affects diffuse irradiance, in particular under a cloudy sky. The corresponding highly cloud dependent
change in diffuse fraction may be estimated from theoretical considerations.
Acknowledgment
This work was supported by the European Commission. It is a part of the work in the project
SATELLIGHT in the JOULE Programme JOR3-CT950041, and we thank our project colleagues for
valuable advice and submission of data.
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decomposition models before and after the eruption of Mt. Pinatubo, June 1991. Solar Energy, 57, 433443.
Olseth, J.A. and Skartveit, A. (1989) Observed and modelled hourly luminous efficacies under arbitrary
cloudiness. SOLAR ENERGY,42, 221-233.
Paltridge , G.W. and Platt, C.M.R. (1976) Radiative processes in meteorology and climatology. Elsevier,
New York.
Perez, R., Seals, R., Zelenka, A. and Ineichen, P. (1990a) Climatic evaluation of models that predict
hourly direct irradiance from hourly global irradiance: Prospects for performance improvements. Solar
Energy, 44, 99-108.
Perez, R., Ineichen, P., Seals, R. and Zelenka, A. (1990b) Making full use of the clearness index for
parameterizing hourly insolation conditions. Solar Energy,45, 111-114.
Perez, R., Ineichen, P., Maxwell, E., Seals, R. and Zelenka, A. (1992) Dynamic global to direct irradiance
conversion models. ASHRAE Transactions Vol. 98, Part 1, 3578, 354-369.
Radiation Yearbooks Nos 1-32 (1965-1996) Radiation observations in Bergen Norway. University of
Bergen, Norway.
Skartveit, A. and Olseth, J.A. (1987) A model for the diffuse fraction of hourly global radiation. Solar
Energy, 38, 271-274.
Skartveit, A. and Olseth, J.A. (1992) The probability density and autocorrelation of short-term global and
10
beam irradiance. Solar Energy,49, 477-487.
Tuft, M.E. (1993) Parameterization of the diffuse fraction of global irradiance. Thesis (In Norwegian),
Geophysical Institute, University of Bergen.
11
0.6
a)
0.20
Overall
0.5
Cloudless
b)
0.15
eqn (3)
eqn (3)
σ3
σ3
0.4
0.3
0.2
0.10
h < 20
0.05
h > 20
0.1
0.0
0.00
0.0
0.4
0.8
1.2
1.6
0.7
0.8
CLEAR SKY INDEX
0.9
1.0
1.1
1.2
CLEAR SKY INDEX
Fig. 1 Group mean values of the variability index σ3 (• or G) plotted vs. clear sky index for overall (a) and
for cloudless sky (b) at Bergen (1965-96). The group mean values in a) are plotted for 5 solar elevation
bins (with almost indistinguishable group mean values of σ3), while a distinction between solar elevation
above/below 20o is made in b). Eqn (3) is also plotted.
0.8
0.6
h=
10
σ3 = 0.0
20
60 30
0.4
5
0.2
90
DIFFUSE FRACTION
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
CLEARNESS INDEX
Fig. 2 Diffuse fraction vs clearness index from our new hourly diffuse fraction model for invariable hours
(σ3 = 0.0) and 6 solar elevations h.
12
0.8
1.0
a)
DIFFUSE FRACTION
DIFFUSE FRACTION
1.0
h = 10
0.6
0.2
0.3
0.4
0.1
0.2
σ3 = 0.0
0.0
0.8
b)
h = 60
0.6
0.4
0.2 0.3
0.2
0.1
σ 3 = 0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
CLEARNESS INDEX
0.2
0.4
0.6
0.8
1.0
CLEARNESS INDEX
Fig. 3 Same as Fig. 2, but for 4 variability indicesF(3) at solar elevation h = 10° (a) and 60° (b).
DIFFUSE FRACTION
1.0
h = 60
0.8
h = 10
0.6
r = 0.60
σ3 = 0.0
0.4
0.2
r = 0.15
0.0
0.0
0.2
0.4
0.6
0.8
1.0
CLEARNESS INDEX
Fig. 4 Same as Fig. 2, but for invariable hours (σ3 = 0.0) at solar elevations h = 10o and 60o and for
surface albedos r = 0.15 and 0.60.
13
DIFFUSE FRACTION (mod)
0.5
Cloudless
0.4
0.3
S&O (old)
S&O (new)*
0.2
S&O (new)
C&S
0.1
0.1
0.2
0.3
0.4
0.5
DIFFUSE FRACTION (obs)
Fig. 5 Modelled vs. observed averages of hourly diffuse fraction for 103 cloudless days at Bergen (196596) for 5 solar elevation bins (<10o, 10-20o, 20-30o, 30-40o, >40o). For each bin, average diffuse fractions
from our old model [S&O (old)], from our new model with bin average σ3 [S&O (new)*] and with zero σ3
[S&O (new)], and from the model of Camps and Soler (1992) are plotted against their observed
o
counterparts (for the C&S model, the lowest solar elevation bin comprises solar elevations < 17
).
0.8
0.6
0.4
S & O (new)
0.2
h < 10
0.0
1.0
DIFFUSE FRACTION
1.0
DIFFUSE FRACTION
DIFFUSE FRACTION
1.0
0.8
0.6
0.4
0.2
S & O (new)
20 < h < 30
0.0
0.0
0.2
0.4
0.6
0.8
CLEARNESS INDEX
1.0
0.8
0.6
0.4
S & O (new)
0.2
h > 40
0.0
0.0
0.2
0.4
0.6
0.8
CLEARNESS INDEX
1.0
0.0
0.2
0.4
0.6
0.8
1.0
CLEARNESS INDEX
Fig. 6 Group mean values (X) of observed diffuse fraction vs clearness index for nearly invariable hours
(σ3 < 0.03) for 3 solar elevation (h) bins at all 4 stations of the verification sample collectively. Curves for
the new model (with σ3 = 0.02) are drawn for the lowest and highest group mean solar elevation within
each solar elevation group.
14
0.5
h < 10
0.4
0.3
0.2
0.2
0.3
0.4
0.5
0.6
b) S & O (old)
DIFFUSE FRACTION (mod)
a) Erbs et al.
DIFFUSE FRACTION (mod)
DIFFUSE FRACTION (mod)
0.6
0.5
0.4
0.3
0.2
0.6
0.2
DIFFUSE FRACTION (obs)
0.3
0.4
0.5
0.6
c) S & O (new)
0.5
0.4
0.3
0.2
0.6
0.2
DIFFUSE FRACTION (obs)
0.3
0.4
0.5
0.6
DIFFUSE FRACTION (obs)
a)
600
400
200
0
(114)
0
200
400
600
800
800
b)
600
400
(125)
MODELLED NORMAL BEAM
IRRADIANCE (Wm-2)
800
(117)
"OBSERVED" NORMAL BEAM
IRRADIANCE (Wm-2)
Fig. 7 Modelled (mod) vs. observed (obs) group mean diffuse fractions for the 4 stations of the
verification sample grouped according to solar elevation (5 bins) and σ3 (4 bins) for high clearness index
groups (0.7-0.8 for solar elevation h > 10o, and 0.6-0.7 for solar elevation h < 10o). Modelled data are
obtained from group mean values of input data by the model of Erbs et al. (1982), and by our old and new
model. A distinction is made between solar elevation bins h above/below 10o.
200
(114)
0
0
OBSERVED NORMAL BEAM
IRRADIANCE (Wm-2)
200
400
600
800
OBSERVED NORMAL BEAM
IRRADIANCE (Wm-2)
Fig. 8 a) Hourly "observed" (global - diffuse) vs. observed (Eppley NIP) normal incidence beam
irradiance for solar elevations h < 10o at Bergen (1991-96). b) Normal incidence beam, obtained from
global irradiance and our new diffuse fraction model, plotted vs. the same NIP data. Overall averages are
printed on the respective axes in both cases. In b) the three highest values obtained by the diffuse fraction
model of Erbs et al. (1982) are also plotted (E).
15
Table 1. Mean Bias Deviation (MBD = modelled - observed) and Root Mean Square Deviation (RMSD)
of hourly diffuse fraction for the Erbs et al. (1982) model, the Maxwell (1987) model, the Perez et al.
(1992) model, and our old (Skartveit and Olseth, 1987) and new model (see text). Deviations are given for
the four stations of the test sample (for each one and for all four stations collectively [All 4]), for Bergen 1
(1965-96) and for Bergen 2 (1991-96). Values are given for solar elevation h > 10o and h <10o, and for all
4 stations collectively values are even given for clearness index k > 0.7. Number of hours (N) and
observed average diffuse fraction (Obs) are also given.
Station
N
Obs
Lisbon
h > 10o
Lyon
h > 10o
Garston
h > 10o
Gävle
h > 10o
All 4
h > 10o
All 4
h < 10o
All 4
k > 0.7
4701
.500
3459
.665
3084
.745
2719
.647
13963
.624
2310
.772
2177
.220
75269
.723
2210
.829
14390
.715
429
.809
Bergen 1
h > 10o
Bergen 1
h < 10o
Bergen 2
h > 10o
Bergen 2
h < 10o
Erbs
Maxwell
SO old
Perez
SO new
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
.006
.100
.036
.113
.013
.096
-.024
.109
.009
.104
.082
.180
-.027
.095
-.022
.109
.024
.107
-.007
.103
-.060
.131
-.015
.112
-.002
.187
-.017
.100
.024
.094
.050
.111
.020
.089
-.009
.111
.023
.101
.016
.181
.034
.091
-.007
.095
.038
.103
.008
.095
-.020
.109
.005
.100
.062
.177
-.002
.077
.005
.086
.039
.099
.011
.088
-.022
.104
.009
.094
.031
.165
-.002
.067
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
.000
.102
.061
.149
.009
.101
.066
.144
-.023
.098
-.033
.124
-.011
.092
-.024
.103
.010
.093
-.013
.112
.012
.092
-.005
.090
.002
.087
.001
.103
.011
.096
.017
.092
.001
.084
-.014
.105
.009
.083
-.010
.085
16
Table 2. Same as Table 1, but for hourly diffuse irradiance.
Station
N
Lisbon
h > 10o
Lyon
h > 10o
Garston
h > 10o
Gävle
h > 10o
All 4
h > 10o
All 4
h < 10o
All 4
k > 0.7
4701
Obs
(Wm-2)
167
3459
154
3084
160
2719
148
13963
159
2310
33
2177
148
75269
144
2210
34
14390
143
429
35
Bergen 1
h > 10o
Bergen 1
h < 10o
Bergen 2
h > 10o
Bergen 2
h < 10o
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
Erbs
(Wm-2)
-4
47
7
41
-6
38
-10
42
-3
43
3
10
-15
57
Maxwell
(Wm-2)
-4
52
11
47
-8
35
-19
45
-4
46
-3
11
-4
68
SO old
(Wm-2)
13
46
19
46
1
30
-1
40
9
42
-1
10
23
60
Perez
(Wm-2)
-1
44
15
43
-3
32
-6
38
2
40
1
9
0
49
SO new
(Wm-2)
2
38
14
38
-2
30
-5
36
3
36
-1
8
-2
42
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
-4
40
5
12
0
38
5
12
-9
39
-3
10
-5
37
-2
9
4
37
-1
8
7
37
0
7
0
33
0
7
3
35
1
7
0
32
-1
8
2
32
-1
7
17
Table 3. Same as Table 1, but for hourly normal incidence beam irradiance (estimated from global and
diffuse irradiance). For Bergen 2 (1991-96), beam irradiance is also measured directly by an Eppley NIP.
For this period, observed beam irradiance and MBD and RMSD compared to these observations are given
in parentheses.
Station
N
Lisbon
h > 10o
Lyon
h > 10o
Garston
h > 10o
Gävle
h > 10o
All 4
h > 10o
All 4
h < 10o
All 4
k > 0.7
4701
Obs
(Wm-2)
446
3459
267
3084
188
2719
312
13963
318
2310
128
2177
781
75269
232
2210
101
14390
236
(236)
117
(114)
Bergen 1
h > 10o
Bergen 1
h < 10o
Bergen 2
h > 10o
Bergen 2
h < 10o
429
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
Erbs
(Wm-2)
0
78
-20
77
4
62
17
85
-1
76
-29
90
27
96
MBD
RMSD
MBD
RMSD
MBD
RMSD
MBD
RMSD
1
75
-34
86
-6 ( -5)
73 (80)
-38 (-35)
87 ( 88)
18
Maxwell
(Wm-2)
21
89
-12
77
19
68
46
101
17
84
21
91
18
101
SO old
(Wm-2)
-18
74
-32
77
-1
52
3
87
-14
73
5
86
-33
91
Perez
(Wm-2)
9
74
-22
69
8
58
14
82
2
71
-20
80
2
76
SO new
(Wm-2)
-2
65
-24
66
5
51
13
77
-3
65
-3
74
2
67
19
-7
1
0
73
68
61
59
22
9
0
9
73
62
56
57
10 (11) -14 (-13) -7 (-6)
-4 (-4)
68 (72) 67 (73) 66 (69) 58 (63)
17 (20)
3 ( 7) -11 (-8 )
8 (12)
63 (70) 52 (54) 54 (52) 49 (55)